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Theorem oncard 7848
Description: A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
oncard  |-  ( E. x  A  =  (
card `  x )  <->  A  =  ( card `  A
) )
Distinct variable group:    x, A

Proof of Theorem oncard
StepHypRef Expression
1 id 21 . . . 4  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  x )
)
2 fveq2 5729 . . . . 5  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  ( card `  x ) ) )
3 cardidm 7847 . . . . 5  |-  ( card `  ( card `  x
) )  =  (
card `  x )
42, 3syl6eq 2485 . . . 4  |-  ( A  =  ( card `  x
)  ->  ( card `  A )  =  (
card `  x )
)
51, 4eqtr4d 2472 . . 3  |-  ( A  =  ( card `  x
)  ->  A  =  ( card `  A )
)
65exlimiv 1645 . 2  |-  ( E. x  A  =  (
card `  x )  ->  A  =  ( card `  A ) )
7 fvex 5743 . . . 4  |-  ( card `  A )  e.  _V
8 eleq1 2497 . . . 4  |-  ( A  =  ( card `  A
)  ->  ( A  e.  _V  <->  ( card `  A
)  e.  _V )
)
97, 8mpbiri 226 . . 3  |-  ( A  =  ( card `  A
)  ->  A  e.  _V )
10 fveq2 5729 . . . . 5  |-  ( x  =  A  ->  ( card `  x )  =  ( card `  A
) )
1110eqeq2d 2448 . . . 4  |-  ( x  =  A  ->  ( A  =  ( card `  x )  <->  A  =  ( card `  A )
) )
1211spcegv 3038 . . 3  |-  ( A  e.  _V  ->  ( A  =  ( card `  A )  ->  E. x  A  =  ( card `  x ) ) )
139, 12mpcom 35 . 2  |-  ( A  =  ( card `  A
)  ->  E. x  A  =  ( card `  x ) )
146, 13impbii 182 1  |-  ( E. x  A  =  (
card `  x )  <->  A  =  ( card `  A
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2957   ` cfv 5455   cardccrd 7823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-er 6906  df-en 7111  df-card 7827
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